Let's break down the given polynomial expression [tex]\( P(x) = -40x^2 - 100x + 27,500 \)[/tex] and analyze its components to answer the question:
1. Constant Term in the Polynomial Expression:
- The polynomial given is [tex]\( P(x) = -40x^2 - 100x + 27,500 \)[/tex].
- The constant term here is [tex]\( 27,500 \)[/tex].
- Since this constant term represents a value independent of [tex]\( x \)[/tex], in the context of the problem, it denotes the daily earnings without any price increases.
2. The Binomial [tex]\((500 - 20x)\)[/tex]:
- This represents the number of tickets sold.
- Initially, 500 tickets are sold at the base price.
- For each [tex]\( \$2 \)[/tex] increase in price (represented by [tex]\( x \)[/tex]), 20 fewer tickets are sold.
- So, the term [tex]\((500 - 20x)\)[/tex] directly adjusts the number of tickets sold according to the price increase.
Using this understanding, we can complete the sentences:
1. The constant of the polynomial expression represents the daily earnings with 0 price increases.
2. The binomial [tex]\((500 - 20x)\)[/tex] is a factor of the polynomial expression and represents the number of tickets sold.
Thus, the completed sentences are:
- The constant of the polynomial expression represents the daily earnings with 0 price increases.
- The binomial [tex]\((500-20x)\)[/tex] is a factor of the polynomial expression and represents the number of tickets sold.
